A geodesic feedback law to decouple the full and reduced attitude

Abstract

This paper presents a novel approach to the problem of almost global attitude stabilization. The reduced attitude is steered along a geodesic path on the n − 1-sphere. Meanwhile, the full attitude is stabilized on SO(n). This action, essentially two maneuvers in sequel, is fused into one smooth motion. Our algorithm is useful in applications where stabilization of the reduced attitude takes precedence over stabilization of the full attitude. A two parameter feedback gain affords further trade-offs between the full and reduced attitude convergence speed. The closed loop kinematics on SO(3) are solved for the states as functions of time and the initial conditions, providing precise knowledge of the transient dynamics. The exact solutions also help us to characterize the asymptotic behavior of the system such as establishing the region of attraction by straightforward evaluation of limits. The geometric flavor of these ideas is illustrated by a numerical example. the second category. The closed-loop kinematics on SO(3) are solved for the states as functions of time and the initial conditions, providing precise knowledge of the workings of the transient dynamics.Recent work on the problem of finding exact solutions to closed-loop systems on SO(n) includes [13,14]. Related but somewhat different problems are addressed in [6,16,22]. Earlier work [12] by the authors is strongly related but also underdeveloped; its scope is limited to the case of SO(3). This paper concerns a generalization of the equations studied in [12,14]. The results of [14] is also generalized in [13], partly towards application in model-predictive control and sampled control systems and without focus on the behavior of the reduced attitude. The work [15] addresses the problem of continuous actuation under discrete-time sampling. The exact solutions provide an alternative to the zero-order hold technique. The algorithm alternates in a fashion that is continuous in time between the closed-loop and open-loop versions of a single control law. The feedback law proposed in this paper can also be used in such applications by virtue of the exact solutions. PreliminariesLet A, B ∈ C n×n . The spectrum of A is written as σ(A). Denote the transpose of A by A ⊤ and the complex conjugate by A * . The inner product is defined by A, B = tr(A ⊤ B) and the Frobenius norm by A F = A, A 1 2 .

Trajectory Tracking for Quadrotors with Attitude Control on $\mathcal{S}^{2}\times \mathcal{S}^{1}$

“…At the time of writing we found similar decoupling approaches for the attitude control of quadcopters in the literature (see [20]- [23]). Our approach differs from previous work in the parametrization we propose, which we believe provides a more natural way to define the rotation about the thrust vector.…”

“…otherwise. (23) Since v(0) = −v(0), then ξ(0) = −1, and since ξ(t) monotonically increases, v(t) = −v(t), for every t ≥ 0. Then ( 23) implies (22), concluding the proof.…”

Geometric Reduced-Attitude Control of Fixed-Wing UAVs

“…The idea of separately controlling reduced attitude and another variable that is decoupled from the reduced-attitude vector is not new. In [64], the reduced attitude is steered along a geodesic path, while the full attitude is stabilized. In [65,66], the attitude control of a quadrotor is decoupled into thrust-vector control on S 2 , and control of the angle of rotation about the thrust vector.…”

Dynamic interpolation for obstacle avoidance on Riemannian manifolds

“…The main advantage of this approach is that it can be used to design global motions for many systems of practical interest where obstacle avoidance is necessary. The results of this work can be applied to a wide range of problems in systems and control such as spacecraft docking [24], quantum control [18], control of quadrotor UAVs [33], multi-agent systems [34], DNA structures [22], control of marine cables/rods [10] and constrained under-actuated spacecraft [2], [3], [38] among others.…”